Measure and Integration
Fall 2015 - Fun Page
Week 2 Worksheet -- "Due" Thursday, September 24
- Read sections 2.1, 2.2 and the first three pages of 2.3
- Write out the proof that any countable set has measure 0.
- Find an open interval cover of the interval (0,1] that has no finite subcover. Is this a counterexample to Heine-Borel?
- Prove a more general version of the Heine-Borel Theorem we proved in class to show that every open interval cover of a closed and boundd set has a finite subcover.
- The Cantor Set, C_{10} can be constructed using Mathematical Induction. The first step of this inductive construction is: "Delete the center 8/10 of [0,1], leaving two disjoint congruent intervals each of length 1/10."
Your task (should you decide to accept it) is to write careful (and accurate!!) directions for the inductive step of this construction. That is, first make an appropriate Inductive Assumption. Then show how to carry out the next step using this Inductive Assumption .
- Prove that the Cantor Set, C_{10}, is uncountable and has measure zero.
- Let C_{3}={x in [0,1]: x has only 0's and 2's in its base 3 decimal
expansion}. Write directions for inductively constructing C_{3} and then
show C_{3} is a null set.
- Learn the proof that a countable union of null sets is null.
- Suppose that for each n, L_{n} is a length where 0< L_{n}<1 and that
Σ L_{n} = M<1. Now, use an inductive construction similar to that of
the one you did for C_{3} and C_{10} to construct a "Cantor Set" of measure
1-M.
- Write (and understand!) a complete proof that the outer measure of a closed
interval is its length. (Theorem 2.3)
- Prove that if two sets differ by a null set, then they have the same outer measure.
(This is Exercise 2.4 on page 25 in the book)
- Determine the smallest σ field,
BiggyPairs, which contains all pairs of rationals.
That is, every set of the form {r_{1}, r_{2}}
(where r_{1} and r_{2} are rationals) is in this σ field (or σ algebra) and BiggyPairs
has nothing extraneous.
Theorems and Definitions to Know by ♥!
- Definitions
- What does it mean for a set A to be a null set?
- Define the outermeasure of a set A.
- What does it mean for a set function to be monotone?
- What does it mean for a set function to be countably subadditive ?
- What does it mean for a collection of sets A to be closed under countable unions (or
complementation, or
countable intersections, or finite unions, or ...)?
- What does it mean for a collection of sets A to be a σ field ?
- Theorem Every countable set is null.
- Theorem If N_{i} is null for i=1,2,..., then U_{i=1}^{infinity}
N_{i} is null.
- Theorem Outer measure is both monotone and countably subadditive.
- TheoremThe outermeasure of an interval is its length.